Math 6510 • Differentiable Manifolds

Let \(X=S^1\vee S^1\), with basepoint \(*\) at the point where the two circles
are glued, and with \(a,b:S^1\to X\) the generators of \(\pi_1(X,*)\) corresponding
to the identity maps onto each circle in the wedge sum. Construct a covering
space \(p:(E,e)\to(X,*)\) such that \(p_*\pi_1(E,e)\) is the subgroup of
\(\pi_1(X,*)\) generated by \(a^2\), \(ab^{-1}\), \(ab\). Justify that your answer works
in two ways:

\begin{enumerate}
\item[(a)] using covering space theory; and
\item[(b)] using the van Kampen theorem.
\end{enumerate}